Representation and poly-time approximation for pressure of $\mathbb{Z}^2$ lattice models in the non-uniqueness region

TL;DR

This paper introduces a new pressure representation theorem for 2D lattice models, enabling efficient approximation algorithms for various models in non-uniqueness regions, with broad temperature applicability.

Contribution

It presents a novel pressure representation theorem and demonstrates its use in developing efficient algorithms for approximating pressure in 2D lattice models, especially in non-uniqueness regions.

Findings

Efficient algorithms for pressure approximation in 2D ferromagnetic Potts models.

Applicability of results to all inverse temperatures except the critical point for Potts.

Extension of approximation methods to certain subcritical and supercritical regions of Widom-Rowlinson and hard-core models.

Abstract

We develop a new pressure representation theorem for nearest-neighbour Gibbs interactions and apply this to obtain the existence of efficient algorithms for approximating the pressure in the $2$-dimensional ferromagnetic Potts, multi-type Widom-Rowlinson and hard-core models. For Potts, our results apply to every inverse temperature but the critical. For Widom-Rowlinson and hard-core, they apply to certain subsets of both the subcritical and supercritical regions. The main novelty of our work is in the latter.